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Pacific Journal of Mathematics THE DENSITY TOPOLOGY F RANKLIN D. TALL Vol. 62, No. 1 January 1976 PACIFIC JOURNAL OF MATHEMATICS Vol. 62, No. 1, 1976 THE DENSITY TOPOLOGY FRANKLIN D . TALL The density topology on the real line is a strengthening of the usual Euclidean topology which is intimately connected with the measure-theoretic structure of the reals. The purpose of this note is to treat the density topology from a modern topological viewpoint. 1. Introduction. A sprinkling of papers has been published by analysts on the density topology. We summarize the important results from the literature, and contribute some new ones. These mainly concern the characterization of certain subspaces, and consideration of cardinal invariants. Many of the topics we touch upon can be treated in more general measure-theoretic structures than the real line, but this does not appear to be particularly fruitful topologically. The organization of this paper is as follows: in §2, results stated explicitly or inherent in the literature are given; in §3, some new results are obtained, especially concerning various subspaces of X ; in §4, the implications for X of various set-theoretic hypotheses are examined. 2. Definitions a n d results from t h e literature. DEFINITION 2.1. A measurable set E C R has density d at x if 2h exists and equals d. Different authors use slightly different definitions of density but they all agree in case d = 1 and therefore determine the same topology. Denote by φ(E), {x GR: d(x, E) = 1}. Let A ~ B mean Λ Δ S (the symmetric difference of A and B) is a nullset (i.e. has measure zero). 2.2. (See e.g. [12].) Let A be measurable. Then (1) φ(A)~A, (2) ifA-B, then φ(A) = φ{B\ (3) φ(0) = 0 andφ(R) = R, (4) φ(AΠB)=φ(A)Γ)φ(B), (5) if A C β, then φ(A) C φ(B). THEOREM 275 276 FRANKLIN D. TALL THEOREM 2.3. (See e.g. [2].) The family of all measurable sets E such that φ(E)D E is a topology on R, henceforth denoted by (X, 3~) or just X if confusion is unlikely. Clearly SΓ is stronger than the usual topology. 2.4 [2]. X is not normal; however X is Tz\, in fact given disjoint sets F and K, one closed in X, the other in the Euclidean topology, there exists a continuous function f: X—»[0,1] such that f~ι(0) = F and THEOREM We shall later give a number of different proofs that X is not normal. That X is T$ is proved in [2] using a consequence in [19] of the important Lusin-Menchoff Theorem: THEOREM 2.5. Suppose (in the Euclidean topology) that Fis closed, B is Borel, F C B, and for every x E F, d(x, B)= 1. Then there is a perfect set P, F C P C B, such that for every x E F, d(x,P) = l. THEOREM 2.6 [14]. (1) The Borel subsets of X are precisely the measurable sets. (2) Every Borel subset ofX is a Gδ, in fact the intersection (or union) of an open Fσ and a closed G δ , namely any E= (E Π φ(E)) U(E- φ{E)\ and similarly for X - E. (3) Every regular open set is a Euclidean Fσ8. THEOREM equivalent: (1) Y (2) Y (3) Y (4) Y is is is is 2.7. The following conditions on a subset Y of X are a nullset, nowhere dense, first category, closed discrete. The equivalence of the first three is shown in [12], as well as that nowhere dense sets are closed, and hence closed discrete. If a closed discrete set were not a nullset, it would include a nonmeasurable set. But nonmeasurable sets are not closed. For a proof of 2.7 in a more general context, see [10]. THEOREM 2.8. RO(X), the regular open algebra ofX, is the reduced measure algebra $ of measurable sets modulo nullsets. Moreover φ(A) = A if and only if A is regular open. The former is easily seen, since RO(X) can be characterized as open sets modulo first category sets [4]. The latter is proved in [12]. THE DENSITY TOPOLOGY 277 THEOREM 2.9. X satisfies the countable chain condition; indeed X has property (K): every uncountable collection of open sets has an uncountable subcollection such that each pair meet. This follows from the corresponding result for measure algebras [6]. 2.10. X is connected [3]. However, there is a T^ enlargement £f of 3~ which satisfies 2.7 inter alia, which is extremally disconnected and the Stone-Cech compactification of which coincides with the Stone space of $ [14]. THEOREM THEOREM 2.11. X is neither separable nor first countable, but is hereditarily Baire (every subspace of X is Baire). This follows immediately from the facts that countable sets are nowhere dense and that first category sets are nowhere dense and closed. 2.12. The continuous real-valued functions of X are of Baire class 1 and hence are continuous in the Euclidean topology except on a set of Euclidean first category [3]. THEOREM The continuous real-valued functions on X are in fact the "approximately continuous" functions. There is an extensive literature concerning these which does not however consider their topological aspects. See Goffman's papers for some references. One may conclude various things about X from the fact that it has a weaker separable metric topology. For example, THEOREM 2.13. X has a countable point-separating open cover and a regular Gδ-diagonaί Both properties are possessed by the Euclidean topology and are preserved by strengthening a topology. X possesses a variety of completeness properties. For definitions and proofs see [18]. THEOREM 2.14. pseudocomplete. X 3. New results. of the above results. is cocompact, strongly a-favorable, and We now move on to some simple applications THEOREM 3.1. Every Y C X is the union of a CCC (countable chain condition) set and a closed discrete set. 278 FRANKLIN D. TALL Proof. Y = (Y Π int Ϋ) U (Y Π ( Ϋ - int Ϋ)). The first term is CCC since that property is inherited by open sets and by dense sets. The second is a subset of a nowhere dense set, and hence is closed discrete. COROLLARY 3.2. Every discrete subspace of X is closed. Proof. Using the Theorem, we see that a discrete Y is the union of a countable (hence closed) set and a closed set. COROLLARY 3.3. X is hereditarily subparacompact (every subspace has the property that open covers have σ-discrete closed refinements). Proof. It suffices to prove CCC subspaces Y of X are subparacompact. Given an open cover 0 of Y, refine it by an open cover °U such that the closures of elements of °U refine 0. Let {Cn}n<ω be a maximal disjoint collection of open sets, each included in an element of °U. Then Y- Un<ωCn is nowhere dense in Y, hence in X, hence is closed discrete. A σ-discrete closed refinement of ϋ can then_be obtained by taking the points of Y — U n < ω C n as one level, and the Cn's as the other levels. DEFINITION 3.4. A space is K-compact if every closed discrete subset of it has cardinality < K. A space is (σ-) metacompact if every open cover has a (σ-) point-finite open refinement. A space is collectionwise Hausdorff if for each closed discrete subset Y there exist pairwise disjoint open sets, each containing exactly one element of Y. THEOREM 3.5. (1) If Y C X is Hrcompact, Y is hereditarily Lindelόf (2) IfYQX is either collectionsise Hausdorff or σ- metacompact, Y is the union of a hereditarily Lindelόf set and a closed discrete set. (3) X is neither collectionwise Hausdorff nor σ-metacompact. Proof. It is easily seen that subparacompact N r compact spaces are Lindelόf. Since closed subsets of X are G δ 's, Lindelόf subsets are hereditarily Lindelόf. If Y is collectionwise Hausdorff, it is the union of an Mj-compact set and a closed discrete set. In [17] it is shown that CCC Baire σ-metacompact spaces are Lindelόf, which completes the proof of the second part, and also shows X is not σ-metacompact. X is clearly not collectionwise Hausdorff. THEOREM 3.6. (1) Countably compact subsets of X are finite. (2) 2H°-compact subsets of X either are nonmeasurable cardinality < 2H°. or have THE DENSITY TOPOLOGY 279 The first is because countable sets are closed discrete, the second because every measurable set of power 2*° includes a nullset of power 2M°. Our next theorem provides a proof of the nonnormality of X using very little information about its measure-theoretic structure, in contrast to other proofs [2], [14]. 3.7. Let Y be a normal topological space such that 2\ Then Y is K-compact. THEOREM \RO(Y)\< COROLLARY 3.8. (1) X is not normal, H (2) if Y C X is normal, Y is the union of a 2 °-compact set and a closed discrete set, H Hϊ (3) // 2 ° < 2 and Y C X is normal, Y is the union of a hereditarily Lindelόf set and a closed discrete set. We first prove the Corollary. To see that X is not normal, we note N that | J R O ( X ) | = 2*°, while X is not 2 °-compact. To prove the second part, we note as before that Y = (Y Π int Ϋ) U (Y Π (Ϋ - int Ϋ)). Surely RO(int Y) C RO(X). On the other hand, Y is dense in int Y, and it is easily verified that in general, for a regular space Z, Y dense in Z implies \RO(Y)\ = \RO(Z)\. Thus \RO(Y Hint Ϋ)\^2"°, so YΠintΫ is K Nl 2 °-compact, and if 2"°<2 , Nrcompact and therefore hereditarily Lindelof. Theorem 3.7 provides a useful method for showing many spaces not to be normal. The idea is due to B. Sapirovskiϊ [15]. To prove it, suppose Y is not K-compact but is normal. Let Z be a closed discrete subspace of Y of cardinality K. Let K be any subset of Z. By normality, there exists an open set UKD K such that Uκ Π (Z - K) = 0. Int Uκ is regular open and includes K. It is not difficult to establish that K->int Uκ is a one-one map from the power set of Z into RO(Y). Another way of proving the nonnormality of X is to again observe that X is not 2M°-compact, and that there are only 2*° real-valued continuous functions on X, not enough to provide sufficiently many Urysohn functions. The fact that there are only 2H° real-valued continuous functions on X follows from the fact that they are all of Baire class 1, i.e. limits of sequences of ordinary continuous functions. The proof of nonnormality of X in [2] yields further information: disjoint Euclidean-dense sets cannot be separated by a Urysohn function. In particular, the rationals and {rV2: r rational} cannot be. On the other hand, since X is regular and countable sets are closed, it is not difficult to show that disjoint countable sets have disjoint open sets about them. I have been unable to determine whether disjoint closed sets, one of which is countable, have disjoint open sets about them. 280 FRANKLIN D. TALL Analogous results to those for normality hold for countable paracompactness, thanks to the following result, which can be proved by an analysis of [1]. THEOREM IRO (Y) I ^ K. 3.9. Let Y be a countably paracompact space such that Then Y is K -compact. COROLLARY 3.10. (1) X is not countably paracompact, (2) if Y C X is countably paracompact, Y is the union of a 2"°-compact set and a closed discrete set, (3) // 2*° = Hi and Y C X is countably paracompact, Y is the union of a hereditarily Lindelδf set and a closed discrete set. In view of the well-known analogies between measure and category [12], it is natural to ask whether there is a category analogue of the density topology. If this question is formulated precisely in a natural way, the answer is no. THEOREM 3.11. There is no topology if on the real line such that Y = (R, Sf) has the following properties (1) Y is regular, (2) RO(Y) = $}', the reduced Borel algebra of Euclidean Borel sets modulo Euclidean first category sets. (3) if U is Euclidean open and N is Euclidean first category, then Proof We first recall some facts about Boolean algebras (see e.g. [4]). If si is a Boolean algebra, define a partial order on P = si - {0} by a ^ b if a Λ b = a. A subset D of P is dense if for each a EP there is a dED such that d g a. For any space Z, the g in RO(Z) is just inclusion. A π-base for a space Z is a dense subset of the inclusion ordering on the nonempty open sets. Thus, if Z is regular, there are π-bases of regular open sets and these are exactly the dense subsets of RO{Z) (more precisely RO(Z)-{0}). Finally, we recall from [4] that Sδ; has a countable dense subset and is atomless (no minimal elements in the partial order). Returning to the Theorem, we see that Y has a countable π-base and is therefore separable. But this contradicts (3). (3) may be replaced by (3') every first category subset of Y is nowhere dense in Y. The point is that, since £8' is atomless, Y has no isolated points, so countable sets are first category. THE DENSITY TOPOLOGY 281 4. S e t t h e o r y . There has lately been considerable interest in the question of the existence of hereditarily Lindelof nonseparable regular spaces [13]. A Souslin space is one, and Hajnal and Juhasz [5] construct one, assuming the continuum hypothesis. A much simpler example than theirs, also assuming the continuum hypothesis, is due to H. E. White [18], who, unaware of the problem, failed to make the (trivial) observation that the hereditarily Lindelof subspace of the density topology he constructed is nonseparable. We shall elaborate on the example here. A Sierpinski set is a set of reals which has countable intersection with every nullset. 4.1 [16]. The continuum hypothesis implies the existence of an uncountable Sierpinski set. LEMMA THEOREM 4.2. Y C X is hereditarily Lindelof if and only if Y is a Sierpinski set. COROLLARY 4.3. The continuum hypothesis implies the existence of a hereditarily Lindelof nonseparable, regular Baire space. The corollary is immediate. To prove the theorem, observe that if Y is hereditarily Lindelof, then every closed discrete subspace is countable. Conversely, if Y is a Sierpinski set, then every nowhere dense subset of Y is countable. Therefore, by 3.1, Y is the union of a CCC set Yλ and a countable set Y2. LEMMA 4.4. A space is hereditarily Lindelof if and only if it is CCC and nowhere dense subsets are Lindelof The proof is left to the reader. Y then is the union of a hereditarily Lindelof set and a countable set, and so is hereditarily Lindelof. It is set-theoretic folklore that there are models of set theory in which 2*°>Hι and there are uncountable Sierpinski sets, and that there are models of set theory in which there are no uncountable Sierpinski sets. Thus THEOREM 4.5. It is consistent with the axioms of set theory that 2"° > Hι and there exists a hereditarily Lindelof nonseparable regular space. THEOREM 4.6. It is consistent with the axioms of set theory that the only hereditarily Lindelof subspaces of X are the countable ones. 282 FRANKLIN D. TALL It is perhaps of interest that an uncountable Sierpinski set may be used to construct a hereditarily Lindelof nonseparable subspace of βN - N. First we note that in Scheinberg's extremally disconnected strengthening of the density topology [14], uncountable Sierpinski sets are again hereditarily Lindelof nonseparable. Scheinberg's space has the Stone space of the reduced measure algebra for its Stone-Cech compactification. Kunen [8] has proven that this Stone space is embedded in βN-N. We next prove some more consistency results. DEFINITION 4.7. A space is metalindelδf if every open cover has a point-countable open refinement. A space has caliber Hi if every point-countable open cover is countable. THEOREM 4.8. The assertions that X is not metalindelόf and that X has caliber Mi are consistent with and independent of the axioms of set theory. Proofs. Consistency is provided by Martin's Axiom plus 2"°>M1, M independence by the continuum hypothesis. Martin's Axiom plus 2 ° > Hλ implies the union of Hλ nullsets is a nullset [11]. A CCC space in which the union of Hx nowhere dense sets is first category, has caliber H} [17]. X is not Lindelof, so if it has caliber Hi it is not metalindelόf. By arguing as in 3.5, one can in fact prove that if the union of Hi nullsets is a nullset then every metalindelόf subset of X is the union of a hereditarily Lindelof set and a closed discrete set. One can prove that the continuum hypothesis implies X does not have caliber Mi by the same methods used to establish this result for the Stone space of the reduced measure algebra in [9]. However it will also follow immediately once we get X metalindelόf, which is a consequence of 4.9. If every set of power < 2H° has measure zero, then every open cover of X has a point — < 2H° open refinement. THEOREM Proof. Let X = {xa}a<2"0. Let °U = {Uβ}β<2^ be an open cover of X. For each α, let f(a) be the least β such that xa E Uβ. Let Vβ = Uβ-{xa:a<β and f(a)έβ}. Then V = {Vβ}β<2*o is the desired refinement. We next consider the cardinal invariants of X. For definitions, see [7], Since X has a closed discrete subspace of cardinality 2"% we have THE DENSITY TOPOLOGY THEOREM 4.10. 283 The spread, weight, and Lindelδf number of X are all 2H\ Since countable sets are closed, it is easy to see that 4.11. The density, tightness, τr~ weight, and character ofX are all uncountable but ^ 2*°. THEOREM Similarly, 4.12. Suppose every set of reals of power < 2M° has measThen the density, tightness, π-weight, and character are all 2M°. THEOREM ure zero. Thus Martin's Axiom or the continuum hypothesis settles the question. For any space Y, it is known that d(Y)g π(Y), t(Y)^χ(Y), π(Y)^d(Y)-χ(Y). We shall show that for the density topology, d(X)^t(X) and hence π(X)^χ(X). This follows from measuretheoretic characterizations of the density and tightness. THEOREM 4.13. (1) d(X) is the least cardinal K such that there is a subset of X of cardinality K with outer measure 1. (2) t(X) is the least cardinal K such that every nonnullset includes a subset of power ^ K with the same outer measure. COROLLARY 4.14. d(X) ^ t(X), ττ(X) S χ(X). Proof It is clear that d and t are respectively at least as big as the cardinals defined on the right. A set of outer measure 1 is clearly dense. Similarly, if every nonnullset Y include^a subset Z of p o w e r s κ_ with the same outer measure, then if x E Y, either x E Y or x E Z Hence t(x, Y)^κ. The next few words are directed to an audience versed in set theory. In the model obtained by adjoining M2 random reals to a model of the continuum hypothesis, it is well-known that 2*° = M2 and that the reals of the ground model have outer measure 1 in the extension. K. Kunen pointed out to the author that in fact in this model every nonnullset includes a subset of power Mi with the same outer measure. On the other hand, he noted that by adjoining Mi random reals to a model of Martin's Axiom plus 2*° > Mi, one obtains a model in which there is a set of power Mi with outer measure 1, and yet there is a set of power > Mi (namely the reals of the ground model) which does not include any subset of power Mt with the same outer measure. Thus THEOREM 4.15. It is c o n s i s t e n t t h a t d { X ) = t ( X ) = H X < 2*°. 284 FRANKLIN D. TALL THEOREM 4.16. It is consistent that d(X)< t(X). By the remarks at the end of §3, the question of whether the 7r-weίght of X can be less than continuum translates into asking whether there can be a dense subset of the reduced measure algebra of cardinality less than continuum. In response to the author's question, R. M. Solovay and K. Kunen both proved that in the model obtained by adjoining K2 Sacks reals to a model of the continuum hypothesis, there is such a dense set of power Hλ. Kunen's proof also established that in that model the character of X is Hx. Thus T H E O R E M 4.17. It is consistent that K π ( X ) = χ ( X ) = H{< 2 °. In conclusion, I should like to thank the referee for his helpful comments. REFERENCES 1. W. Fleissner, CH implies that countably paracompact, separable Moore spaces are normal, preprint. 2. C. GofFman, C. J. Neugebauer, and T. Nishiura, Density topology and approximate continuity, Duke Math. J., 28 (1961), 497-506. 3. C. Goffman and D. 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